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...r mathematics at Christ Church college, Oxford for 27 years. 'A tangled tale' merges his two talents as storyteller and mathematician. It consists of 10 short humorous stories which present one or more mathematical problems. The 10 'knots' as they are called, were first published in 'The Monthly Packet' magazine between April 1880 and March 1885, where readers were invited...

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...Se confunda adesea paradoxul cu exercl- siul plat si plicticos al reabilitarii truismelor. Nu se poate nega fapsll ca pentru a intoarce 10- curile comune isl trebule curaj.Nimic mai rlscant decit a lucra cu banalltatea.si totuGi,paradoxul bine facut atinge pragul filosoflel.Devine 0 forrna penetranta de cllnoastere.Valoarea de excep1tie a paradoxului care di ...

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... 8. On The 97-Th, The 98-Th, And The 99th -- Smarandache's Problems 50 -- 9. On The 100-Th, The 101-St, And The 102nd -- Smarandache's Problems 57 -- 10. On The Ll7-Th Smarandache's Problem 62 -- Ll. On The Ll8-Th Smarandache's Problem 64 -- 12. On The 125-Th Smarandache's Problem 66 -- 13. On The I26-Th Smarandache's Problem 68 -- 14. On The 62-Nd Smarandache's Problem 71...

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...Preface 5 Chapter One n-VECTOR SPACES OF TYPE II AND THEIR PROPERTIES 7 1.1 n-fields 7 1.2 n-vector Spaces of Type II 10 Chapter Two n-INNER PRODUCT SPACES OF TYPE II 161 Chapter Three SUGGESTED PROBLEMS 195 FURTHER READING 221 INDEX 225 ABOUT THE AUTHORS 229...

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...ysis of the number of relations of the type S(n)- S(n+1)=S(n+2)-S(n+3) for n<10 8 where S(n) is the Smarandache function leads to the plausible conc... ...which can be represented as a sum of consecutive integer is n ways. 10 There may be many positive integers which can be represented ... ...19 5 40 5 670 5 1342 5 12 7 27 7 477 7 957 7 1 14 7 15 217 15 441 15 6 10 1 20 150 21 310 21 15 6 12 12 79 35 175 35 52 2 51 4 21 64 12 105 ... ...on Npp p j j = 12 12 αα α ... we have d(N)=(α 1 +1)(α 2 +1)…(α j +1) or (10) n+1=(α 1 +1)(α 2 +1)…(α j +1) The first step is therefore ... ... to the exponents in ascending order starting with p 1 =3. Every factor in (10) corresponds to a different prime even if there are factors which are... ... R(n)= 3 4 ⋅5 2 ⋅7 2 ⋅11 2 ⋅13=156080925 When n is even it is seen from (10) that α 1 , α 2 , … α j must all be even. In other words the smalles... ... 5 45 3 2 ⋅5 6 729 3 6 7 105 3⋅5⋅7 8 225 3 2 5 2 9 405 3 4 5 10 59049 3 10 11 315 3 2 5⋅7 12 531441 3 12 References: 1.... ... 20 k/j 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 2 ↑ 3 ... ... F 4 =65537 while F 5 =641⋅6700417 as well as F 6 , F 7 , F 8 , F 9 , F 10 and F 11 are all known to be composite. 23 Table ...

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...osition of the result is determined only by the remainder upon division by 10 of the terminal digit raised to that power. Taking each in turn and det... ...s the powers take o to any power is zero to any power is one 21 modulo 10 = 2 22 modulo 10 = 4 2 3 modulo 10 = 8 24 modulo 10 = 6 2 5 modulo... ...0 = 2 22 modulo 10 = 4 2 3 modulo 10 = 8 24 modulo 10 = 6 2 5 modulo 10 = 2 2 6 modulo 10 = 4 6 etc. 3 1 modulo 10 = 3 3 2 modulo 10 = ... ...dulo 10 = 4 6 etc. 3 1 modulo 10 = 3 3 2 modulo 10 = 9 3 3 modulo 10 = 7 3 4 modulo 10 = I 3 5 modulo 10 = 3 3 6 modulo 10 = 9 etc. 4... ... modulo 10 = I 3 5 modulo 10 = 3 3 6 modulo 10 = 9 etc. 4 1 modulo 10 = 4 4 2 modulo 10 = 6 4 3 modulo 10 = 4 44 modulo 10 = 6 etc. 51 ... ... 4 2 modulo 10 = 6 4 3 modulo 10 = 4 44 modulo 10 = 6 etc. 51 modulo 10 = 5 52 modulo 10 = 5 etc. 6 1 modulo 10 = 6 6 2 modulo 10 = 6 etc... ...lo 10 = 5 etc. 6 1 modulo 10 = 6 6 2 modulo 10 = 6 etc. 7 1 modulo 10 = 7 7 2 modulo 10 = 9 7 3 modulo 10 = 3 7 4 modulo 10 = I 7 5 mo... ...7 7 2 modulo 10 = 9 7 3 modulo 10 = 3 7 4 modulo 10 = I 7 5 modulo 10 = 7 etc. 8 1 modulo 10 = 8 8 2 modulo 10 = 4 8 3 modulo 10 = 2 8... .... 8 1 modulo 10 = 8 8 2 modulo 10 = 4 8 3 modulo 10 = 2 8 4 modulo 10 = 6 8 s modulo 10 = 8 etc. 9 1 modulo 10 = 9 9 2 modulo 10 = I 9...

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...1 Binary Relation 7 1.2 Mappings 9 1.3 Semigroup and Smarandache Semigroups 10 2. Elementary Properties of Groups 2.1 Definition of a Gr... ...ll mappings of S to itself. The number of elements in S(3) is 27 = 3 3 . 10 Example 1.2.2: Let S = (1, 2, 3, ... , n). The set of all mappings... ...ned so as to make the semigroup a monoid or vice versa. Example 1.3.6: Let Z 10 = {0, 1,2, ... , 9}. (Z 10 , ×) is a monoid under the usual mult... ...1,2, ... , 9}. (Z 10 , ×) is a monoid under the usual multiplication '×' modulo 10. Thus, every monoid is obviously a semigroup with the identity. T... ...... , 6} and B = {1, 6}. Thus, Z 7 is a Smarandache semigroup. Example 1.3.10: Let S 2×2 = {(a ij ) |a ij ∈ Z 2 = {0, 1}} be the set of all 2... ...position of maps. PROBLEM 5: Find all subgroup of S 2×2 given in example 1.3.10. With this we end this chapter suggesting the reader the follow... ...many examples of S-semigroups. Example 4.2.1 Let Z 12 = {0, 1, 2, ... , 9, 10, 11} be the semigroup under multiplication modulo 12. Clearly, Z ... ..., 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24} that is Z 25 \ {0, 5, 10, 15, 20} = B is also a subgroup under multiplication modulo 25. T... ...igroup but Z 25 is not a Smarandache 34 cyclic semigroup as Z 25 \ {0, 5, 10, 15, 20} = B is a proper subset of Z 25 which forms an abelian g...

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... A B = 301 411 211 210 110 001 ⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥ ∪ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ − ⎣ ⎦⎣ ⎦ 10 is a 3 × 3 square bimatrix. A' B = 1100 2001 2001 1010 0003 0 ... ...⎢⎥ ⎢⎥ ⎢⎥ ⎪⎪ ⎪⎪ ⎜⎟ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ −− − ⎣⎦ ⎣⎦ ⎣⎦ ⎣⎦ ⎩⎭ ⎩⎭ ⎝⎠ = 52 4 4 50 18 10 43 −− ⎡⎤ ⎡⎤ ⎢⎥ ⎢⎥ −∪ −− ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ − ⎣⎦ ⎣⎦ . ii. Let A B = (... ...1.2.5: Let A B = 30 11 12 02 ⎡ ⎤⎡ ⎤ ∪ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ and B B = 20 21 10 00 − − ⎡ ⎤⎡ ⎤ ∪ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ be two 2 × 2 square bimatrices. ... ... 20 101 11 010 ⎡⎤ ⎡ ⎤ ⎢⎥ ∪ ⎢ ⎥ ⎢⎥ ⎣ ⎦ ⎢⎥ ⎣⎦ 21 and B B = 101 03 001 10 100 ⎡⎤ ⎡ ⎤ ⎢⎥ ∪ ⎢ ⎥ ⎢⎥ ⎣ ⎦ ⎢⎥ ⎣⎦ A B B B = 301101 2003 101001 1... ... B = 32 201 14 021 03 ⎡⎤ ⎡ ⎤ ⎢⎥ −∪ ⎢ ⎥ ⎢⎥ ⎣ ⎦ ⎢⎥ ⎣⎦ and B B = 20 011 10 201 01 ⎡ ⎤ ⎡⎤ ⎢ ⎥ ∪ ⎢⎥ ⎢ ⎥ − ⎣⎦ ⎢ ⎥ − ⎣ ⎦ . The product A B B B is ... ... ⎦ . The product A B B B is defined as A B B B = 32 20 011 201 14 10 201 021 03 01 ⎡⎤⎡ ⎤ ⎡⎤ ⎡⎤ ⎢⎥⎢ ⎥ −∪ ⎢⎥ ⎢⎥ ⎢⎥⎢ ⎥ − ⎣⎦ ⎣⎦ ⎢⎥⎢ ⎥ − ⎣⎦⎣ ⎦ ... ...ustrated by the following example. Example 1.2.9: Let A B = 32 0100 10 1000 31 0101 ⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥ ∪ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ and B B = 011 ... ...000 31 0101 ⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥ ∪ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ and B B = 011 012 100 10 1 011 010 ⎡ ⎤ ⎢ ⎥ − ⎡⎤ ⎢ ⎥ ∪ ⎢⎥ ⎢ ⎥ −− ⎣⎦ ⎢ ⎥ ⎣ ⎦ . Now both A B a... ...rices. Clearly then product A B B B is defined, for A B B B = 32 012 10 10 1 31 ⎡⎤ ⎡ ⎤ ⎢⎥ ⎢ ⎥ ⎢⎥ − − ⎣ ⎦ ⎢⎥ ⎣⎦ ∪ 011 0100 100 1000 011 0101 0...

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...er multiplication modulo 9 with unit. Example 1.1.2: S = {0, 2, 4, 6, 8, 10} is a semigroup of finite order, under multiplication modulo 12. S ha... ... semigroup under multiplication. Clearly I = {0, 7} and J = {0, 2, 4, 6, 8, 10, 12} are ideals of Z 14. Example 1.1.9: Let X = {1, 2, 3, … , n} ... ... map is the identity element under composition of mappings. Example 1.1.10: Let S(3) be the semigroup of order 27, (which is for n = 3 described ... ...is a group. (with respect to the same induced operation). DEFINITION 1.1.10: Let S be a S semigroup. If every proper subset of A in S, which is a... ...s a S-semigroup then obviously S is a Smarandache commutative semigroup. 10 Let S be a S-semigroup, ο(S) = number of elements in S that is the ... ...primes p is a S- semigroup. For more about S-semigroups one can refer [9,10,11,18]. PROBLEMS: 1. For the semigroup S 3 ×3 = {(a ij ) / a ij... ...roup homomorphism between g = 〈g/g 6 = 1 〉 and S 4 ? Prove or disprove. 10. Does there exist a group homomorphism between G = 〈g/g 11 = 1 〉 and ... ...ce with respect to the restriction of ∩ and ∪ of L onto S. DEFINITION 1.2.10: Let L and M be any two lattices. A mapping f: L → M is called a 1... ... Figure 1.2.5 0 1 c b a Figure 1.2.6 16 Example 1.2.10: This lattice is non-distributive left for th...

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...  +         = k p E p p k E p k E p k E p k E k e log 3 2 ) ( L 10 And using the lemma we have () [] n p n p n p p E n n n n... ... k) 2 3 1 3 7 1 4 5 1 5 31 1 6 7 1 7 127 1 8 17 1 9 73 1 10 31 1 11 89 1 12 13 1 13 8191 1 14 127 1 15 151 1 16 257 1 ... ...Smarandache Factors and Reverse Factors”, Smarandache Notions Journal, Vol. 10, www.gallup.unm.edu/~smarandache/ 2. Chris Caldwell, The Prime Page...

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...CEPTS AND PROPERTIES 1.1 Fuzzy Relational Equations and their properties 10 1.2 Properties of Fuzzy Relations 16 1.3 Fuzzy compatibility re... ...57 1.9 Neural fuzzy relational system with a new learning algorithm 63 1.10 Unattainable Solution of FRE 67 1.11 Specificity shift in sol... ... and its application to predict the behaviour to a motor-drive system 197 2.10 Application of genetic algorithm to problems in chemical indust... ...or largest solution 268 4.9 Solving NRE with a linear objective function 269 4.10 Some properties of minimal solution for NRE 270 4.11 Application ... ...bibliography. The second chapter deals with the applications of FRE. This has 10 sections: we elaborately give the applications of FRE in flow rat... ... have been trying to explore the problem and develop solution procedures [1, 4, 10-12, 18, 34, 52, 75-80, 82, 108, 111]. The max-min composition is... ...ng the names of all researchers whose research papers have been used. 10 1.1 Binary Fuzzy Relation and their properties It is well k... ...J j∈ sup w i (b, a j ) 9. w i (b, J j∈ inf a j ) = J j∈ inf w i (b, a j ) 10. i [a, w i (a, b)] ≤ b. Prove for the fuzzy relations P(X, Y),... ...x o A = b, 0 ≤ x i ≤ 1 and Minimize ∑ = = m 1 i i i x " c ' ' Z (10) Subject to x o A = b, 0 ≤ x i ≤ 1 where c’ = (c’ 1...

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... )} ( ) ( 2 { ) ( I Z a I Z a P I Z ∈ + = = {(2n+1) + (2n+1) I ⏐ n ∈ Z}. 10 Clearly P I Z ) ( is a group. For P = 2Z (I ) serves as the additi... ...c, each such graph is denoted by K n K 1 K 2 K 3 K 4 K 5 FIGURE: 2.1.10 22 A simple graph with n vertices can have atmost ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ... ...G if it is not properly contained in another clique of G. DEFINITION 2.1.10: Deletion of vertices and edges in a graph. Let G be a graph. X be pr... ...consists of just a single tree. v 2 v 3 v 4 v 5 v 6 v 1 v 7 v 8 v 9 v 10 B 1 B 2 B 3 B 4 B 5 FIGURE: 2.3.7 v 2 v 3 v 4 v 1 B... ... v 4 v 1 B 1 v 4 v 5 v 6 B 2 v 6 v 7 v 8 B 3 v 8 v 9 B 5 v 7 v 10 B 4 FIGURE: 2.3.8 38 There is only one tree with one point, two ... ...HEOREM 2.6.9: If cl (G) is Hamiltonian then G is Hamiltonian. THEOREM 2.6.10: If cl (G) is complete then G is Hamiltonian. FIGURE: 2.6.6 49 ... ... is a cubic simple graph with a cut-edge then χ' (G) = 4. DEFINITION 2.7.10: A snark is a cyclically 4-edge connected cubic graph of girth at lea... ...ween the nodes is given in the form of directed graph given in Figure: 2.8.10. From the above signed directed graph, we obtain a connection matr... ... matrix is a 8 × 8 matrix. Thus we have E = [A y ] 8 ×8 FIGURE: 2.8.10 C 1 C 2 C 3 C 8 C 7 C 4 C 5 C 6 -1 -1 -1 -1 +1 +1 +1 -1 ...

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........................................ 9 Evelyn Brady ?Safeguard? .....................................................................................10 Jenna Boyle ?Last Night? .......................................................................................12 Kate F. Van Schaick (Judge?s Selection for Best Poem) ?Ruskie? ............................. 14 Michael Rie...

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...or College, 38 -- Absence: A Farewell Ode on Quitting School for Jesus College, Cambridge, 39 -- Happiness, 40 -- A Wish: Written in Jesus Wood, Feb. 10, 1792, 43 -- An Ode in the Manner of Anacreon, 44 -- To Disappointment, 45 -- A Fragment Found in a Lecture-Room, 46 -- Ode, 47 -- A Lover?s Complaint to his Mistress, 49 -- With Fielding?s ??Amelia??, 50 -- Written After ...

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... . 40 Happiness . . . . . . . . . . . 43 A Wish: Written in Jesus Wood, Feb. 10, 1792 . . . . . . . . . . . . . 44 An Ode in the Manner of Anacreon ... ...ish in the blaze while I the blaze admire. 1 by fiercer Gill outdone. MS - 10 - The Nose Coleridge: Poems To the Muse 1789 Tho’ no bold fligh... ...!’’’ - 42 - Happiness Coleridge: Poems A Wish: Written in Jesus Wood, Feb. 10, 1792 1792 Lo! through the dusky silence of the groves, Thro’ v... ...nt limbs Compose with icy hand! - 43 - A Wish: Written in Jesus Wood, Feb. 10, 1792 Coleridge: Poems An Ode in the Manner of Anacreon 1792 As... ...of Justice meet my way — And told me that her name was HAPPINESS. January 10, 1795 To Robert Southey of Baliol College, Oxford, Author of The ‘‘Re... ... of Trinity House, etc. 1800 [As printed in Morning Post for January 10, 1800.] To the Editor of The Morning Post. MR EDITOR —An ...

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...Ehia kahi iloko o ka 10? He 10 a me na kahi ehia iloko o ka 12? He 10 a me na kahi ehia iloko o ka 13? 14? 16? 19? 15? 18? 17? 11? Ehia na umi iloko o ka 20? iloko o ke 30? 40? 60? 80? 60? 70? 50? 90? 100? Ehia na umi a me na kahi iloko o ka 21...

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...they don’t have the best toys in the world. What Best of Freshman Writing 10 was important was that they had each other. I haven’t seen any of my chi... ...: 44-45. Drummond, Tammerlin. “Battling the Columbine Copycats.” Time (May 10, 1999): 29. Gibbs, Nancy. “Special “Report the Littleton Massacre.” Time... ...r in 1973 that the idea was resurrected and would eventually become rule 6.10. The rule states: “Any league may elect to use the Designated Hitter Rul... ...lenges TV Networks Shy of Cash.” New York Times 19 Sep. 2001, late ed.: B5 10 Oct. 2001 <http:// proquest.umi.com>. Beatty, Sally. “Overloaded America...

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...Preface. 5 -- About the cover 9 -- Welcome from Campus President, DrRaul Ramirez. 10 -- Registrants/Attendees. 11 -- Keynotes. 12 -- The light barrier: its construction and demolition. 14 -- Jim Malmberg. 17 -- DrDavid Iadevaia. 25 -- DrLarissa BBorrisova & Dmitri DRabounski. 29 -- DrFlorentin Smarandache....

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...…………………………..…… 9 Welcome from Campus President, Dr. Raul Ramirez ………….. 10 Registrants/Attendees……………………………………………. 11 Keynotes…………………………………………... ... the speed of a beam of light is independent of observer motion. 10 Welcome from Campus President, Dr. Raul Ramirez Welcome to all ... ...g at one gravity (980 cm/sec 2 ) for one year will have a velocity of sec / 10 05 . 3 365 24 60 60 980 10 cm x = ⋅ ⋅ ⋅ ⋅ exceeding the velocity of li... ... 3 365 24 60 60 980 10 cm x = ⋅ ⋅ ⋅ ⋅ exceeding the velocity of light sec / 10 00 . 3 10 cm x . To see if this is true one needs to solve for th... ... (2) Now let g = 980 cm/sec and c = sec / 10 00 . 3 10 cm x and make use of the fact that a year is 7 10 1536 . 3... ...a year is 7 10 1536 . 3 365 24 60 60 x = ⋅ ⋅ ⋅ seconds and a light year is 10 10 3 365 24 60 60 x ⋅ ⋅ ⋅ ⋅ cm which equals 17 10 4608 . 9 x cm. to w... ...τ τ g Sinh g d g Cosh d u t (10) ∫∫ = = = ) ( 1 ) ( 1 τ τ τ τ g Cosh g d g Sinh d u x ... ...t?" respectively, http://www.physlink.com/dcforum/general_astro/3.html. 10. Illingworth, Valerie, editor, "Dictionary of Physics", The Penguin, ... ... belong, as it is claimed that neutrinos do mutate to the other families. 10. Dark material of galaxies and other systems. This list contains ele...

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...n SCBF(n) is defined as: The sum of composite numbers between the smallest prime factor of n and the largest prime factor of n. For example, SCBF(14)=10, since 2£7 = 14 and the sum of the composites between 2 and 7 is: 4 + 6 = 10. In reference [2]: A number n is called simple number if the product of its proper divisors is less than or equal to n. Let A denotes set of all ...